Clue :

Let AP = 1 unit

Then PB = 2 units

(because AP/PB = 1/2, which means that AP : PB = 1 : 2 and PB = 2AP)

AB = AP + PB = 1 unit + 2 units = 3 units

Since AB = AC, AC = 3 units as well


Since AQ/QC = 1/2,

AQ : QC = 1 : 2 and QC = 2AQ

Since AC = AQ + QC, ,

AQ : QC : AC = 1 : 2 : 3

Since AC = 3 units, this means that AQ = 1unit, QC = 2 units


Therefore AP = AQ and PB = QC

Clue 2 :

Draw a line from A to X.

Compare the areas of :

①APX and BPX
②AQX and CQX

What is the ratio of their base lengths? What is the ratio of their heights?

Once you have figured out the ratio of their areas, then compare areas of :


ABC and APC
ABC and AQB

What is the ratio of their base lengths? What is the ratio of their heights?

Can you deduce the area of APC and AQB from here?


After you have found them, think about :

Which triangles make up APC?
Which triangles make up AQB?